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Metamath Proof Explorer

Theorem rankidn

Description: A relationship between the rank function and the cumulative hierarchy of sets function R1 . (Contributed by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion rankidn ⊢ A ∈ ⋃ R1 On → ¬ A ∈ R1 ⁡ rank ⁡ A

Proof

Step Hyp Ref Expression
1 eqid ⊢ rank ⁡ A = rank ⁡ A
2 rankr1c ⊢ A ∈ ⋃ R1 On → rank ⁡ A = rank ⁡ A ↔ ¬ A ∈ R1 ⁡ rank ⁡ A ∧ A ∈ R1 ⁡ suc ⁡ rank ⁡ A
3 1 2 mpbii ⊢ A ∈ ⋃ R1 On → ¬ A ∈ R1 ⁡ rank ⁡ A ∧ A ∈ R1 ⁡ suc ⁡ rank ⁡ A
4 3 simpld ⊢ A ∈ ⋃ R1 On → ¬ A ∈ R1 ⁡ rank ⁡ A